QA 

218) 


v^aspc. 


IN  MEMOmAM. 
George  Davidson 


University  of  California 


lA 


DIRECT  AND  GENERAL  METHOD" 


FINDING  THE  APPROXIMATE  ¥ALUEg 


THE  ItEAL  KOOTS  OF  NOMERICAL  EQUATIONS 


ANY  DEGREE  OF  ACCURACY, 


By  d.  WJNICHOUSON,  A.  M., 


I*rcsi(ii;m  ;uiii  I'rofcssor  nt  M;ilhem;itics  in  the  Louisiana  State  University  and  Agricultural  and  Mecl-.anical  College, 

And  author  of  Nicholson's  Mathematical  Series. 
Baton  Rouge,  La. 


NKW  (JULliANS: 
V.  V.  UAXSELL  &  BHO.,  PUBLISHERS. 

ISftl. 


|)j^|^ 


i 


DIRECT  AND  GENERAL  METHOD 


OF 


Finding  the  Approximate  Values 


OF 


THE  REAL  ROOTS  OF  NUMERICAL  EOOATIONS 


ANY   DEGREE   OF  ACCURACY. 


BY  J.  W.  NICHOLSON,  A.  M., 


Presultnt  and  I'rofessor  of  MaUiematics  in  the  Louisiana  State  University  and  Agricultural  and  Mechanical  College,  Baton  Rouge,  i. 

And  author  of  Nicliolson's  Mathematical  Series. 


NEW  ORLEANS; 
K.  F.  UANSELL  &  BRO,  PUBLISHERS. 

1891. 


I'tUNTED  BV 

1/.  Graham  &  Son 
new  okleans. 


Coi'YKiGHT  i8gi,  iiv  J.  W".  Nicholson. 
AU  rights  rrstrvfd. 


INTRODUCTION, 


In  this  small  treatise  I  have  undertaken  the  difficult  task  of  producing  a  direct  and  absolute 
method  of  finding  the  approximate  values  of  all  the  real  roots  of  numerical  equations.  The  prac- 
tical utility  of  the  method,  whether  great  or  small,  has  been  with  me  a  matter  of  secondary 
consideration. 

The  most  celebrated  methods  yet  discovered  of  approximating  the  roots  of  equations  are 
those  of  Lagrange,  Newton  and  Horner;  but  all  these  a.v&  tentative,  since  they  depend  primarily 
on  ^/-/'a/ substitutions  of  numbers  for  the  unknown  quantity.  It  is  believed  that  this  want  of  com- 
pleteness is  for  the  first  time  theoretically  and  practically  overcome  by  the  method  now  proposed. 
Indeed,  judging  from  the  general  solutions  of  cubic  and  biquadratic  equations,  if  it  were  possible 
to  express  the  roots  of  higher  equations  in  terms  of  their  coefficients  by  means  of  radicals,  the 
reduction  of  the  expressions  to  their  simplest  forms  would  be  more  laborious  and  less  direct  than 
thj  process  of  finding  the  roots  as  herein  presented,  especially  when  all  the  roots  are  real. 

This  method  may  be  termed  a  generalization  of  that  of  Newton,  of  which  a  competent  judge 
says,  "It  is  comparatively  easy  of  application  and  rapidly  attains  its  object."  However,  the  gen- 
eralization, if  such  it  may  be  called,  was  effected  by  the  employment  of  principles  and  methods 
which  appear  to  have  formed  no  part  of  Newton's  conception  or  deduction  ;  and  I  trust  the  follow- 
ing statement  in  reference  to  the  matter  may  not  be  inappropriate: 

Several  years  since  I  was  extracting  the  square  and  cube  roots  of  numbers  by  the  principles 
of  inequalities  (see  Nicholson's  Elementary  Algebra,  pp.  212-217),  and  the  idea  occurred  to  me 
that  the  same  principles  might  be  advantageously  applied  to  the  solution  of  numerical  equations. 
That  idea,  though  vague  in  its  inception,  is  the  germ  of  the  present  production. 

In  the  development  of  this  idea,  I  disco  veered  that  "  successive  substitutions  "  tend  to  a  defi- 
nite limit  in  the  intervals  where  the  given  function  is  an  increasing  function  of  x.  Largely  by  the 
application  of  this  princijsle,  without  reference  to  or  even  a  thought  of  Newton's  method,  I  obtained 
the  following  general  formula: 


^t(^x) 


(j)i  —  s)fix)  —  xf'{x)    J     ^' 


The  demonstration  of  this  formula  is  not  essential  to  our  present  purpose,  but  in  reference 
to  it  I  desire  to  state : 

(1)  It,  together  with  the  elementary  formulas  on  which  it  is  based,  affords  the  means  of 
finding,  without  trial  substitutions,  the  approximate  values  of  the  real  roots  of  numerical  equations. 


-^  -»^  -^iJ^UM^^  /"A  O 


Introduction. 


(2)  It  assumes  n  different  forms  or  values  as  «  is  made  equal  to  1,  2,  3, «,  in  succes- 
sion. Each  of  these  forms  is  peculiarly  adapted  to  finding  the  corresponding  root  oi  /\.v)  =  0, 
when  that  root  is  positive. 

(3)  When  «  =  M  it  reduces  to 

^^'^  =  '-r&> ' ; ^'^ 

which  is  the  formula  of  Sir  Isaac  Newton. 

Before  noticing  specially  the  form  which  ^  (.r)  assumes  when  « :=  n,  I  had  prepared  a  paper 
on  the  former  and  its  application  to  the  solution  of  numerical  equations.  But,  having  subse- 
quently observed  that  Newton's  formula  was  one  of  the  n  forms  of  ^(.c),  the  idea  then  occurred 
to  me  that  this  formula  might  be  made  the  means  of  accomplishing  all  I  desired,  by  employing  it 
according  to  the  principles  and  methods  which  led  to  the  discovery  of  the  general  formula  (1). 

Therefore,  since  F(_x)  is  of  a  simpler  form  than  ^  (a;),  although  not  always  so  direct  and 
rapidly  converging,  I  abandoned  my  original  treatise  and  prepared  the  present  pamphlet. 

An  impoi'tant  feature  of  this  method  is  its  applicability  also  to  equations  involving  imaginary 
roots.  By  it  we  determine  "  the  number  and  place  of  the  real  roots"  by  finding  the  roots  them- 
selves. 

In  point  of  practical  utility  it  furnishes  very  simple  and  economic  solutions  of  cubic  and  bi- 
quadratic equations.  The  solutions  of  higher  equations  are  tedious  and  laborious  by  any  method  ; 
yet,  in  many  instances,  the  work  is  simplified  and  abridged  by  the  present  process. 

The  examples,  which  are  quite  numerous  for  so  small  a  treatise,  have  been  selected  with  a 
view  of  presenting  all  the  conditions  and  cases  that  can  possibly  arise,  and  to  illustrate  every  part 
of  the  subject.  I  might  have  chosen  examples  better  adapted  to  a  more  favorable  display  of  the 
method,  but  preferred  to  restrict  the  selection  to  such  as  usually  appear  in  our  best  text-books. 

With  the  hope  that  all  efforts  to  extend  and  perfect  science  by  the  removal  of  limitations,  in- 
dependent of  the  intrinsic  worth  of  their  attainment,  are  duly  appreciated,  and  with  the  belief  that 
the  present  treatise  will  be  awarded  that  meed  of  praise  commensurate  with  its  merits,  the  author 
freely  commits  it  to  a  generous,  intelligent  and  discriminating  public. 


NOTATION   AND   PRINCIPLES. 


NOTATION. 

1,  For  brevity  and  uniformity  we  shall  employ  the  following  notation: 

(  I  )  An  expression  like  (i>^>6ora<;y<fo  signifies  that  the  value  of  y  is  between  that 
of  a  and  /y,  or  that  y  may  have  any  value  in  the  interval  from  a  to  I). 

(2)  An  expression  like     r  \  F  {oc)  \  a     signifies: 

1°.  r  is  to  be  substituted  for  x  in  Fix),  and  the  result  F  (r),  (  :=  r, ,  say),  substituted  for 
x'ln  F  {x),  and  that  result  F  i^i),  (  = '"; ,  say),  substituted  for  x  in  F  {x),  and  so  on. 

2°.  Each  new  result  will  be  nearer  the  value  of  a  than  the  preceding  one,  so  that  there  is 
no  limit  to  the  accuracy  which  may  be  obtained  in  thus  finding  the  approximate  value  of  a.  That 
is,  a  is  the  limit  of    r  [  F  (x)  |  a. 

Hence,  when  r  and  F^x)  are  known,  r  |  F  (■»)  |  d  is,  theoretically,  the  value  of  a,  and 
■practically  a  formula  by  which  that  value  may  be  determined  to  any  degree  of  accuracy.  When 
we  wish  to  express  it  in  the  former  sense,  we  write  simply 

r\F{x)^a (3) 

It  is  our  purpose  to  show  how  such  values  of  every  real  root  ot  any  numerical  equation  may 
be  directly  determined. 

(3)  The  general  equation  of  the  n  th  degree  will  be  denoted  by  f  (,x)  =  0,  and  the  succes- 
sive derivatives  of  f(x)  by/'  {x),  f"  (a;),  /'"  (x),  etc. 

(4)  The  roots  of  f^x)  =  0,  in  descending  order  of  magnitude,  will  be  denoted  by  a, ,  a, , 
a,, ,  etc.  ;  those  of  f  (a?)  =  0  by  aj ,  a'^,  aj ,  etc.  ;  those  of  f"  (a?)  =  0  by  a"  ,  aj' ,  aj  ,  etc. ;  and 
so  on.     Thus,  a^  is  read :   the  third  toot  of  the  fourth  derivative. 

(5)  Imaginary  roots  will  be  denoted  by  i;  thus,  ffli  and  a^  =  i,  is  read:  a,  and  «j  are 
imaginary. 

(6)  The  superior  and  inferior  limits  of  the  roots  oi  f  (x)  =  0  will  be  denoted  by  /,  and  L; 
those  oi  f'  (a;)  =  0,  by  l\  and  ij ;  of/"  {x)  =  0,  by  I"  and  Vi ;  and  so  on.  These  limits  are  deter- 
mined by  the  ordinary  algebraic  methods.  When  the  roots  are  all  positive  we  shall  make  ^  =  — 
the  coefficient  of  a;""',  and  l.^  =  —  the  quotient  of  the  coefficient  of  x°  divided  by  that  of  x. 


DIRECT    AfETriOb    OF    FINDING    THE    APPROXIMATE    VaLu£s 


(7)  The  functions  F {x),  Fi  (x),  F.,  (■*■)>  etc.,  will  always  have  the  following  values: 
Fix)=  X  —fj^.  F,ix)=.x~C^;   F.  (.r)  =  x—^1^.  etc  (4) 

PRINCIPLES. 

2.  If  either  of  the  roots  of  /"(.»)  =  0,  as  a, ,  be  substituted  for  x  in  F  (•<?)  the  result  will  be 
that  root.     That  is,  F (a,)  =  a..     Similarly,  Fi  («!)  =  a\;  and  so  on. 

3.  If  F  {x)  is  an  increasing  function  of  X  when  a  <  x  <  h,  then  P  (rt)  <  P  (^)- 

4.  F  (.)?)  is  an  increasing  function  of  x  at  all  points  where  its  derivative  is  positive.  That 
is,  when  F'  (x)  >  0. 

5.  F  (x)  is  an  increasing  function  of  x  in  the  intervals  where  f{x)  and  f"  (.r)  have  the  same 
sign.     For,  taking  the  derivative  of  the  value  of  F  {x)  we  have: 

F'(x)==.f{x)f"ix)^[rix)y (5) 

Hence,  F'  {x)  >  0  when /(a;)/"  (.«)  >  0. 

6.  Evidently  /  (a?) /"  (x)  >  0,  (1)  when  «,  <  .r  <  Z, ,  (2)  when  ^  <  .f  <  «„ ,  and  (T.) 
when  «,]>»>  ay_i  or  a"_i  >  ;»  >  «, .  Hence,  F  (a;)  is  an  increasing  function  of  x  between  the 
same  limits,  or  in  the  same  intervals. 

7.  Since  li  ])>  «!  ,  and  F  (*)  ^n  increasing  function  of  x  in  the  interval  from  <i,  to  J,  ,  F  (/,) 
>  F  (tti)  ;  but  jf'  (ttj)  =  rt, ;  hence  F  (/,)  >  ffj . 

Again,  since  l.^  <^  a„  we  may  show  in  a  similar  manner  that  F  (Jj)  <C  «» • 


DETERMINATION  OF  THE  SUPERIOR   AND   INFERIOR    ROOTS 

OF  AN   EQUATIOiN. 


8.  The  superior  root  of  f  (.r)  =  0  /s  ?i  |  Jp^  (x). 

For,  substituting  l^  for  *•  in  F  (x),  remembering  that/  (J^)  >  0,   and  /'  (/j)  >  0,  we  have 

By  Art.  7,  F  (/J  >  «i ;  hence  ?j  >  F  (/j)  >  ffj. 


OF    THE    REM,    ROOTS    OF    XU.MERICAI,    EQUATIONS. 


That  is,  if  Zj  be  substituted  for  x  in  F  (.r)  the  result  will  be  in  the  interval  between  ij  and 
a^,  and  if  that  result  be  substituted  for  x  in  F  (x)  it  will  be  still  more  nearly  equal  to  flj  ;  and 
SO  on. 

Hence  (.3),         J^  |  JF  (x)  =  a^ (6) 

9.   The  inferior  root  off{x)  =  0  is  l^   \  F  (x). 

For,   substituting  l^    for  x  in  F  (x),  rememberino;  that  when  »  is  even  f  (^2)   >    0  ^'^^ 
/'  Oi)  <^  0'  ^"^'  vice  versa  when  n  is  odd,  we  have  l^  <  -f'  (^2)- 
By  Art.  7,  ^  (Z^)  <  ff,„ ;   hence,  Z^  <  J?'  (Zj)  <  «,. 
Therefore,  as  before,  v.-e  have  (3) : 

h  I  F(x)=a (7) 


exa.n«f»i-e:s. 


1.  .7;'  —  6  a?  4-  1  —  0. 


Here,  ?,  =  6,  /,  =  i,  and  F  (x)  =  (^illf +1>.     Hence  (6),  (7), 

J     (iX/     0) 


i\  F(x)  \  a,  =  .171  + 
.17  I  "  =.171572  =a. 


&  \  F  (x)\  ai  =  5.83  + 
5.83  I        "  =  5.828427  =  a, 

These  answers  are  correct  to  six  decimal  places. 

2.  aP  ~  G  x''  +  9  x  —  2  —  0. 

Here,  /,  =  6,  i,  =  f  =  .2  +  ,  F^x)  =J(J^IZ^J^+J1.     Hence  (G),  (7), 


6  \Fix)  Ia,=  4.8  + 

5  I         "         =4.2  + 

4  I         "         =  3.7  + 

.3.7  I         "         =3.731'  + 

3.73  I         "         —  3.73205  =  a, 

Tliese  results  are  correct  as  far  as  extended. 


.2     I  JP  (X)  1  «3  =  .264  + 
.26  I  «'  =  .267  + 

.27  I  "  =  .26794  =r  a. 


Note.— In  finding  fli  we  may  use  ^  (x),  see  (  1  ),  instead  of  F  {x)  by  making  n  :^Z  and 
s  =  1 ;  and  we  shall  find  that  the  approximation  is  more  rapid. 


3.  x' 


7  j;  +  7  =  0. 


Here,  '1  =  3,  /, 


I  and  F  (.r)  =  ' 


2  x"  ■ 


'6  X-  —  7 


DIRECT    METHOD    OF    FINDING    THE    APPROXIMATE    VALUES 


3  I  F  (.*•)  1  fl,  =  2.3  —  4  i  F  (x)  I  a.,  =  —  3.1 

2  I         "  =:  1.8  —3  I        "  =—  3.05 

1.8  I  "  =1.7  —  3.05  I         "        "      =  — 3.0189  =ff, 

1.7  I  "  =  1.692 

1.69  I  "  =  1.6920  =  «, 

These  results  are  correct  as  far  as  carried.  It  will  be  observed  that  the  approximation  in 
finding  a,  is  comparatively  slow,  this  is  owing  to  the  fact  that  n.  is  nearly  equal  to  a,  ,  as  we  shall 
subsequently  show.  See  ex.  8.  If  ^  (x),  (1),  is  used  in  place  of  F  (a;)  the  approximation  will 
be  more  rapid. 


^  3  (  (x^  — 4)  .r^  4-  1) 
4  (  (a-2_6)a;  -f  3) 


4.  J7*  —  12  x=  +  12  a;  —  3  =  0. 
Here,  h  =  4,  t,  =  —  4  and  F  {x) 

i:  \  F  {x)  \  a^  =3.3  + 
3  I         "  =  2.87  + 

2.9  I         "  =  2.859  + 

2.85  I         "  =  2.8580  =  a, 

5.  if'  —  10  a^  +  6  a;  +  1  =  0. 


Here,  ?,  =  4,  /,  =  —  4  and  F  (x)  =  i-"^^'''''  ~  ^Zli. 

.        ■'       b  x^  (^x'  _  6)  -{-  6 


-i\F  ix)  I  «, 
—  3.91  I         " 


—  3.91  + 
:  —  3.9073  =  a. 


4  I  F  (.r)  I  a,  =  3  + 
3  I         "  3.056 

3.05  I         "  3.0530  =  aj. 


—  4 

1  F  (.r) 

". 

= 

--  3.4 

—  3.4 

1         " 

= 

—  3.06 

-  3.00 

1         " 

- — 

—  3.0653 

10.  Generally,  when  the  successive  values  do  not  tend  d'rectly  towards  a  definite  limit,  the 
root  sought  is  imaginary.  It  sometimes  happens  that  the  successive  values  jump  from  small  to 
large  numbers,  or  vice  versa,  and  then  tend  towards  a  definite  limit.  In  such  cases  we  are  to  infer 
that  the  root  sought  is  imaginary,  and  the  result  is  another  real  root  of  the  given  equation. 


«.    Find  the  inferior  root  of  x^  —  6  a'^  +  10  a?  —  8  =  0, 

(2  X  —  0)  x^  -\-  8 


Here,  I,  =  .8  and  F  (*) 


(3  X  —  12)  j;  +  10 


The  successive   values  of   .8  |  JF  (-i')  |   a~   are  +   2.2, 
imaginary. 


1.3,   —  .2,   etc.     Hence,   flj  is 


6P    tHE    KKAL    toots    OF    NUMERICAL    EQUATIONS.  ,     9 


DETERMINATION   OF  ALL  THE   REAL   ROOTS. 


PRINCIPLES. 

11.  The  ,9  th  root  oi  f  (x)  =  0  and  the  («  —  l)th  root  of  /"  (x)  =  0  are  both  between 
the  «  th.  and  the  («  —  1)  th  roots  of  /'  (.r)  =  0,  all  the  roots  being  real.  That  is,  a'.  <  a,  < 
«;_! ,  and  «;  <  <■_,   <  ai_i . 

12.  In  the  interval  from  a',  to  a,,  f  (,x)  and  f'  (a;)  always  have  contrary  siorns,  and  in  the 
interval  from  a,  to  a',_,  they  always  have  the  same  sign.  Therefore,  remembering  that  2*'  (.«■)  = 
X  —  (/(•*)  )  -^  (/'  (i'?)  ),  we  have: 

(1)  Whena?_,  <a.,        a';_,  <_  F  (a^_,). 

(2)  When  <_,>«.,         «:■_!>  J' (C_0- 

13.  Since  F  («)  is  an  increasing  function  of  x  in  the  interval  from  a"_,  to  a,,  Art.  C,  we 
have : 

(  I  )  When  <_,  <  a. ,         F  «_,)  <  F  (fl.)  or  <  a, ,  Art.  2. 

(2  )  When  a;'_i  >  «. ,         F  (ai'_i)  >  JF  («.)  or  >  a, ,  Art.  2. 

In  either  case  then,  see  Art.  12,  ^  «'_,)  is  in  the  iriterval  from  <_,  too,.  Similarly, 
r  being  any  number  between  ffi"_,  and  a,,  F  (>')  is  between  r  and  a, .     Hence, 

a':^,  \  F  {X)  =  a. (9) 

14.  Now  let  us  find  a','_i.  Since /"^(o;)  sustains  the  same  relation  to/'"(j?)  that  the  latter 
does  to  f  {x),  we  have : 

ar_2  I  F,  (,<^)  ==  a';_, (10) 

Similarly,  aJL,  |  F^ix)=a':^, (H) 

etc.,  etc.,  etc.,  etc. 

16.  It  will  be  seen  that  the  solution  of  f  (x)  =  0  depends  on  the  solution  off"  (a;)  =  0; 
the  latter  on  that  of  /"  (as)  =  0 ;  and  so  on,  as  follows: 

I.  We  have  (6)  (7),  I,  |  F  {x)  =  a,  ,  L  \  F  (x)  =  a„ (12) 

II.  By  (12),  A-  I  F,  {X)  =  < ,  l\  i  F^  (a?)  =  aS_i. 


10  DIRECT    METHOD    OF    KlNOiNG    THE    APPROXIMATE    VALUlJS 


Now  ill  (9)  make  8  =  2,  and  s  =  n  —  1,  and  we  lia\c: 

«!■  \  Fix)=a,_,  «;;_,  I  F  (.r)  =«„_,. 

Therefore 

l\'  \  F^  (x)  =  a1 ;  a'.;  \  F  (x)  =  n,      \ ^ ^^^ 

'2  i  F.,  (.v)  =  o;;_ 2 ;  u-'_^^  \  F  ix)  =  fl„_ .     j 

III.  In  a  similar  manner  we  obtain 

17  I  F,  (X)  =  aV ;  a\'  \  F^  (a?)  =  «■■  \  a^  \  F  (.r)  =  a,     >  _ 

i'J  I  l*'^(a?)=C-4;  C-,  I  -P2  (•T')=«;;-3;  C-s  I  F{x)^a„^S 

The  process  may  evidently  be  carried  on  indefinitely. 

16.  A  Descending  Solution  of  an  equation  consists  in  finding  the  roots  successively,  begin- 
ning with  the  largest;  and  an  Ascending  Solution  consists  in  finding  the  roots  similarly,  beginning 
with  the  smallest. 

WHEN   THERE  ARE   IMAGINARY    ROOTS. 

17.  In  a  descending  solution  if  all  the  roots  greater  than  a,  have  been  found  to  be  real, 
and  a"_i  \  F  (^x)  =  a,  proves  to  be  imaginary,  we  infer  that  «,  +  i  is  also,  and  hence  proceed  to  find 
the  second  root  below,  viz:  a".^i  \  F  {x)  ==  «,_^2  . 

In  an  ascending  solution  under  similar  circumstances  we  seek  the  second  root  above,  viz : 
o?_3   I  F{,x)  =  a...^. 

18.  Under  the  same  conditions,  if  a"  and  a"^■^  are  also  imaginary,  a"_i  |  F  (:»)  will  ^^  pass 
over"  the  two  imaginary  roots  a,  and  a,  +  ,  and  produce  the  root  «.  1-2  . 

For  under  the  conditions  stated  f  {_x)  f"  (a?)  >  0  in  the  entire  interval  from  a;'_,  to  «,+  3  . 
Hence, 

<-i  \F{x)=a.^^ ■ (15) 

In  general,  if  all  the  roots  from  a,  to  a. +  3,-1,  and  from  «;' to  a"  +  2<-i'  inclusive,  are 
imaginary,  we  have: 

al'_,  I  F{x)=a.+  ^, (IC) 

Thus,  making  «  =  1,  (  =  1,  and  a;'_,  =  /,  ,  we  have 

l,\  Fix)=a, (H) 

And,  when  <  =  2,         k  \  F  ^x)  =  a, (1«) 


6v    THE    RKAL    ROOTS    OK    NUMEiUCAL    EQUATIONS.  11 

19.  Again,  in  an  ascending  solution  under  the  conditions  stated  in  Art.  17,  if  «"_j  and  a".  3 
are  also  imaginary,  «f,_i  |  -F  (x)  will  pass  over  the  imaginary  roots  a,  and  «,_i  and  produce  the 
root  a,_2- 

For  under  these  conditions,  f  (,x)  f"  (j")  >  0  in  the  intei-val  from  a"~i  to  a,.,.     Hence, 

C_i  I  l^(,c)  ^a._2- (19) 

In  general,  if  all  the  roots  from  rt,  to  «.,_3,_  j ,  and  from  a"^^  to  al'_2,_3,  inclusive,  are 
imaginary,  we  have: 

ai-.i  I  F  (,ij)  =  «._,. (20) 

Thus,  making  s  =  1 ,  (  =  1  and  a"_  ^  ==  /^  ,  we  have 

h     I    J'    («)=:«„_, .....(21) 

If  «  =  6,  t  =  1,        a\  I  ^  (.r)  =  fl„ (22) 

20.  The  preceding  principles  apply  also'  to  /"  (a;)  =  0  and  /"'  (.r)  =  0,  to  /"  (.r)  =  0  and 
f"  (,r)  =  0,  and  so  on.  Therefore  we  are  enabled  in  any  given  equation  to  determine  all  the 
real  roots.     The  following  illustrations  will  render  this  fact  very  clear. 

21.  Let/(.r)  =  0  be  an  equation  of  the  fourth  degree. 

We  need  not  consider  the  case  in  which  the  roots  oi  f"  (iC)  =  0,  viz:  fflj' ,  a'^ ,  are  real,  for 
in  that  event  the  application  of  formulas  (13)  is  obviously  direct.  Let  us  suppose,  then,  that  rt^' 
and  aj  =  i. 

If  the  roots  oi  f  (x)  =;  0  are  real,  the  roots  of  /"  {x)  =  0  are  also  ;  hence,  if  the  latter  are 
imaginary  then  two  or  more  of  the  former  are  also. 

(1)  Suppose  a,  and  a^  =  i,  then,  Art.  18,  ^,  |  F  (.»)  will  pass  over  n,  and  a.^  and  give  a^ , 
and  Ij  \  F  (x)  will  give  fl^ . 

(2)  Suppose  a^  and  a,  =  i,  then  i,  |   F  (x)  will  give  a^  ,  and  l^  |  F  (.»)  will  give  a^ . 

(3)  Suppose  tts  and  a,  =  i,  then,  Art.  19,  I2  \  F  («)  will  pass  over  a^  and  a,  and  give  a^  , 
and  li  1  F  (x)  will  give  a, . 

(4)  Suppose  «! ,  fflj  )  «3 1  <»*  —  '1  t'le"  'i  I   ^ (,•"')  ''*"'•  'a   i  -*^  C*^')  ^''^  '^'"^'^  S'^^  *'■ 

22.  Again,  let/'(.r)  ^  0  be  an  equation  of  the  fifth  degree. 

Suppose  a"  and  a'!,  =  i,  then  two  or  more  of  a, ,  a^  ,  «, ,  a,  are  imaginary.  Now  this  case 
is  entirely  similar  to  the  one  which  we  have  just  considered ;  and  the  same  will  be  the  case  if  we 
suppose  «.V  and  a"  =  1. 


12 


DIRECT    METHOD   OK    KlNDlNCi    THE   AH'KOXIMATE    VAl.UkS 


23.  Therefore,  formul:is  (12),  (13),  (14),  etc.,  are  general;  that  is,  the  real  roots  of 
f"  (.t)  =;  0,  together  with  /,  |  F  (J^)  and  l^  \  F^-v),  are  sufficient  to  determine  all  the  real  roots 
of/(.x)  =  0;  similarly,  the  real  roots  of /"(,*•)  r=  0,  together  with  ll'  \  F^  (ar)  and  Z'^  JFj  (.t),  are 
sufficient  to  determine  all  the  real  roots  of /"(a;)  =  0;  and  so  on. 

The  process  is  fully  illustrated  in  the  solution  of  the  following  examples,  in  the  selection  of 
which  an  attempt  has  been  made  to  anticipate  all  the  conditions  and  cases  that  can  possibly  arise. 

24.  When  there  are  imaginary  roots,  and  especially  when  the  given  equation  is  incomplete, 
it  sometimes  happens  that  two  or  more  of  the  results  are  equal  to  each  other.  Now  in  such  cases, 
if  we  know  there  are  no  equal  roots,  which  can  always  be  determined  and  generally  by  inspection, 
we  are  to  infer  that  the  results  which  differ  from  each  other  are  the  real  roots,  and  the  only  real 
roots  of  the  given  equation.     See  Examples  15  and  22. 

25.  In  finding  the  roots  of  /'"  (a;)  =  0,  f"  (a;)  =  0,  etc.,  no  great  accuracy  is  required. 
One  place  of  decimals,  and  often  the  integral  part  of  the  root,  will  suffice,  unless  the  roots  of 
/  (a;)  =  0  are  very  nearly  equal  to  each  other. 


ClTBir  EQUATIONS. 

26.  Let/'(.r)  r=  0  be  an  equation  of  the  third  degree. 
By  formulas  (12)  and  (13),  we  have 


.(23) 


Note. — Evidently  nj' 


-|  of  the  coefficient  of  J'-*. 


EXAMPLES. 


7.  a^  —  9  .^2  +  22  a;  —  11  =  0. 


Here,  /,  =  9,  a'{  =  3,  /,  =  .5,  and  F  (r)  =    (2  a^  —  9)3^^-1-  11^ 

^  (3  a-  —  18)  a;  -f-  22 

Therefore,  9  |  F  (a;)  =  «! ,    3  |  ^(a:)  =  w.^  ,    .5  |   F  (.r)  =  a^  . 


9\F{x)\a,  =  l  + 
7  I         "  =6  + 

6  I         "  =hA  + 


3  I  F  (.r)  I  ttj  =  3.2 


3.2  ! 


3.201G  =rt., 


.5.4  I  J?' (a-)  I  rt,  =  5.17  + 
5.1  I         "  =5.129  + 

5.13  I        "  =5.1284  =  a, 

.5  I  F  (a-)  1  a,  =  .65  + 
.fi5  I         "  =  .669  + 

.66  I         "  =  .6699  c=a„ 


These  answers  are  correct  as  far  as  carried. 


OF     rilK    REAL    ROOTS    OF    NUMKKICAI,    EQUATIONS. 


13 


8.    Find  the  intermediate  root  of  example  3. 

2  J''  —  7 


Cl'  =  U,   :iii(l    F  (.<■)  = 


6  X- 


1  I         "  =  1.25 

1.2  I         "  =  1.32 

9.  .r'  —  6  x'^  +  15  ,r  —  10  =  0. 

I,  =  6,  a'i  =  2,  L  =  .6,  and  F{x)=^  ^^  j;  -  6)  x«  +  10 
'    '  '  ^   '       (3a;— 12)x'+15 

The  successive  values  of  6  ]  J^  (.i)  are  4  -f ,  3  —  ,2  — ,  .6,  .9, 
wc  infer  that  flj  and  a,  ==  i,  and  a~^=-\-l.     See  Art.  10. 


1.3  \  F  {x)  \  a^  ^  1.35 
1.36  I    "      =  1.3568  =  rt^, 


10.  J-'^  —  15  a;  +  21  ^0. 

i,  =  5,  «;•  z=  0,  /,  -^  _  5,  J'  (aO 


2  .r^  — 21 

3  j;2  —  15' 


5  (  F  {X)  ==  2.67247  =  «, 
0  I  FXx)  =  1.76915  =ffj 
b  \  F{x)=,~  4.44162  =  «, 


1.      From  which 


.(21) 


EQUATIONS  OF  THE  FOURTH  DEGREE. 

27.  Let/"(j;)  =  0  be  vn  equation  of  the  fourth  degree,  then  we  have  (12),  (13), 

h  \Fix) =a,  -) 

li'  I  F^ix)  :=.«/■;  «!■  I  Fix)=a,   ! 

I'i  I  F^  (.»•)  =  a.\ ;  «'^  I  1*^  (.r)  =  «,    I    

h  I  1^(0 =«.  3 

Norn. — We  m.^y  evidently  use  «,  and  «,  for  I'i  and  /'/,  ,  rcspc-ctivelv.     It  is  best  to  tind  ((','  and 
aj  by  liic  ordinary  method. 

EXAMPLES. 

11.    Find  the  second  and  third  roots  of  example  4. 

«;'   ::^  1.4  ,    iC  ==  —  1.4,  F  (X)  = 

1.4  1  F  {X)  =  .OOfiOlS  ^  ,ij     I 


3  (  (x^  -  4)  x^  4-  1) 

4  (  (.r'^— 6)   a;  +  3) 

1.4  I  Fix)  ==  .443270  =  a, 


14 


DIRECT    METHOD    OF    FINDING    THE    APPIIOXIMATE    VALUE!> 


12.  x*  —  ir'  —  Sx  +  21  =  0. 

7,  =  4,  /,  =  -  3, 
flV  =  2,  «'i  =      0, 

i  \  F  (X)  =3.6796  =a, 
2  i       "        =  2.2674  =  («, 


13.  x'  —  12  jr'  +  39  J;-^  —  48  X'  +  10  =  0. 
Here,  /,  =  12,  I.,  =  .2,  «i'  =  i.b,  a''  =  1.4. 

12  I  F  (x)  =  7.741  =«,  I 

4.5  I        "      =i  =a^  I 

14.  X*  —  a  n^  -\-  32  .«2  —  77  *■  +  2.5  =  0. 


^    '^       (4  a;  —  12)i'2  —  3 


0  I  F  (.»)  =  J  =  rt, 

3  1       "       =  i  =  a. 


1.4  I  J^  (j;)  =  i  =  a, 

.2  I      "       =  .258  +  =  a, 


Here,  /j  =  5,  l^  =;  .3,  a"  =  i,  a'j  =»'.      Hence,  Art.   23,    the  only  possible   real  roots  are 
5  I  *'(«)  and  .3  j  F  {v). 


6  I  F  (x)  =  2.6180  =  Og  I 

15.  a;*  —  28  x  —  4.5  =  0. 
7,  =  4,  ^2  =  -  2,  «V  =  "i'  =  0,     F  (X)  = 


.3  I  JP  (a;)  =  .3819  =  a, 


3  X*  +  45 


4  07'  —  28 
4  I  JP  (a;)  =  3.4494 ;  0  |  ^  (x)  —  —  1.4494 ;  ^  2  |  ^  (a;)  =  —  1.4494. 

Now,  as  there  are  no  equal  roots,  Art.  24,  the  only  real  roots  are  3.4494  and  —  1.4494, 

16.  x*+  2  X  +  3.75  =  0. 


7,=0,  72=-2,  «i'=«'i=0,         F(x) 


3x*  —  3:lo 

ix^  +  2~' 

0  I  F  (.r)  =  *  and  —  2\F  (x)  =  i. 
.*.  «] ,  a.,,  ti^  and  a,  are  imaofinary. 
17.  x'  —  19  x^  —  23  .r  —  7  =  0. 
/  -  5    7    -  -4     Fix)-  <l^^^^^±-l. 

fl5'  =  1.7,  «v=  —  1.7 

5  I  F  (.«■)  =  4.8977  =  «,     —  1.7  j  ^  (.!■)  =  —  .7124  =  rt, 
1.7  I  ^  (a)  =  —  .5522  =  rt^    ^  4  ]  JP  (.e)  =  —  3.6331  ==  «, 


6f  the  real  roots  of  numerical  equations. 


15 


.(25) 


EQUATIONS  OF  THE  FIFTH  DECREE. 

28.  Let/ (a?)  =  0  be  an  equation  o!  the  fifth  degree,  then  by  (12),  (13),  (14),  we  have 

h\  F(x) =  «,   " 

i;-  I  F^  ix)  ^  rt'/  ;  rtS'  I  F  (x)  =  a^ 

flr  I  F^  (.»•)=«'•. ;  «2  I  Fix)  =  a, 

/■■  I  F^  (.0  =«"  ;  "s  I  Fix)=a, 

h  I  F  (.r) =(h  3 

Note. — In  place  of  l\  and  V^  we  may  write  a^  and  a^  respectively,  and  in  place  of  a'l ,  —  J 
of  the  coefficient  of  .»*  . 

18.  Find  rtj  ,  rt.3  Bnd  a,  in  example  5. 

4  it^  r^^ 5^ 1 

«!■  :=  1.7,  «■■  =0,  «;,'  =  —  1.7,  and  Fix)  =  5  ^2  (^.2  _  6)  +  6* 

1.7  \  F  ix)  =  .87950870  •  =  a^  .         0  \  F  (.»')  ::=  —  .17567479  =  a,. 
_  1.7  I  j^  (a;)  =:  — .69157628. 

19.  x'—ax'+  25  Cf^—iOx^  +  .34  .x;  —  12  =  0. 

(2  a!  — 4.8)  .c'+  4 
''  =  «'  '-^  =  -3'  «-^  =  1-6.  ^3  (^)  -  ^3¥^l,.6)a;+7.o- 

We  shall  use  a'  and  ffj  instead  of  i"  and  l'^  . 

8  I  JF  (.r)  =  3  =  «,,    3  j  1^2  (.r)  =  2.2  ;     2.-2  \  F  ix)  =  2  =  a^  . 
1|F,  (*o=i. 
.3  I  Fix)  =  l=a,. 

Hence,  the  three  real  roots  are  3,  2,  1.     See  Art.  23. 

20.  X'—  11  X*  +  49  x'  —  109  x'  +  120  *  —  50  =  0. 

11  I  Fix)  =  1. 
This  is  evidently  the  least  root;  hence  all  the  others  are  imaginary.     See  (18). 

21.  9  a;'  —  3  .X*  —  26  .1;'  —  1  =  0. 

/j  =  -f  3,  ai'  =  1,  fl'^  =  0,  n^  =  —  .8  and  '3  =  —  2, 


16 


DIRECT    RfETHOI)    OF    FINDING    fHE    APPROXIMATfe    VAi,(jES 


1  I  F  (_x)  =i  =  a^ 
0  I  F (r)  ^i  =  a^ 

22.  a-'^  —  14  a?  —  3  —  0. 


3  I  J'  (a;)  =  4-  1.879385242  =  a, 
—  .8  I  F  {x)——  .347296355  =  a^ 
—  2  I  ^  (a;)  =  —  1.53208888(;  ^  a. 


Here,  i,  =  2,  i^  =  —  2,  a?  =  a"  =  aj  =  0,  JP  (a?)  =  g-^J^u* 

Now,  Art.   24,  as  tliere    are    no    equal    roots  the   only  possible  real  roots  are  2  |  F  (•<), 
—  2\F  (x),  0  i  F  (*■)• 

2  I  J'(a?)  =  1.9+,   0  \  F(,x)  =  -  .21-  ,   —  2  j  JF  (x)  ==  —  1.8  — . 


EQUATIONS  OF  THE  SIXTH  DEGREE. 

29.  Let/(.»)  =0  be  an  equation   of  the  sixth   degree,  then  by  (12),  (13),  (14),  we  have 

h\  F(x)  =a, 

l'{  I  F^  (0?)  =.  al ;     a\  \  F(x)  =  a^ 
aY  I  F^  ix)  =  a'i  ■     a\^  \  F  (x)  =  a^ 


a';  I  F^  (X)  ^  a'i  ;  a'i  \  F  (a)  =  a, 
ll  I  F.  ix)  =  «V  ;  «V  I  -F  (X)  =  a, 
?2  I  -P    (•»)  =  «0 


•(26) 


EXAMPLES. 

88.  .■»«  —  8  ,r<  +  15  a'i'  +  2  a;  —  1  =  0 . 


Here,  /i  =  4,  l^  =  —  4,  «;'=  1.68, 
a^  =  .59  ;    aV  =^  —  .59  ;    a\'=z~  1.68 

4  I  J' (a;)  =2.128419  =  Oj 

1.7  \  F(x)=  1.879385  =  a^ 

.6  I  ^(a;)  =    .201639  =  as 


Fix)  = 


5  a^  —  24  a;'  +  15  a^'  f  1 

6  x"  —  32  a;''  4-  30  x  +  2    " 


—  .6  I  J' (a;)  z=  —    .347296  =  «< 

—  1.7  I  J'  (a;)  =  —  1.532088  =  «j 

—  4  \  F(x)=  —  2.330059  =  a, 


24.  a^  _  14  ar'  4-  70  a;*  —  164  x^  +  203  x'—  126  a;  +  22  =  0. 

?,  =  14,  Ji'  =  fli,  flr=  3.2,  «!/=  1.4  ;    l^^a,,  l,  =  -17  +  (.2,  say). 

Fffr}  —    22—3;"  (203  —  a;  (328  —  x  (210  —  a;  (56  —  5  a;)))) 
'       126  — a;  (406  —  a;  (492 —a;  (280 —x  (70  —  6  a;))))* 


F,  ix) 


203  —  x^  (420  —  a;  (280  —  45  x)) 
492  —  X  (840  —  ar  (420  —  60  x))  ' 


OF    THE    REAL    ROOTS    OF    NUMERICAL    EQUATIONS. 


17 


14  i  F  ix) =  6.236068  =  d, 

6  I  F.,  {X)  =  4.8  +  ;    4.8  ,  F  (.»)  =  3.732050  =  a, 
3.2  !   F^  (.%•)  =  2.4  +  ;    2.4  |  F  (,t)  =  1.763932  =  n, 

1.4     j     F2    (^)    ==   t  =   «•  =   «4 

.26     Fiix)  =zi  =i.  =  (t, 

.2  I  F  ix) =     .267949  =  nr, 

25.  2  .r'  —  25  ,tr  +  13  .1;  —  3  =  0. 

Here,  /i^=  3,  l,  =  —  2,  two  of  the  roots  of/"  (_x)  =  0  are  1.3  +  and  0,  and  the  other  two 
are  imaginary.     Hence,  the  only  possible  real  roots  are  (Art.  23)  : 


3  I  F{x)  =  2.2469 
1.3  I  F  (.r)  =    .5549 


0  I  F(x)  =       .2675 
—  2  I  F  ix)  =  —  .8019 


EqUATIONS  OF  THE  SEVENTH  DEGREE. 

30.  Let/"(.i')  =  0  be  an  equation  .of  the  seventh  degree,  then  by  the  preceding  formulas 

the  roots  are 

h  I  Fix) =a. 


V;  I  JPs  (x)  =  «;' 

/r  i  jp',  {x)  =  flr 

«r'  I  F,  (a;)  =  a'^ 
V;  I  F,  (a;)  =  a'; 
/'^  I  Fi  {x)  =  rti' 
h\  F  ix) 


F(.x) 


«r   I   F^  (X)  =  «i' ; 

«2  I  F._  ix) 
<  I  F2  ix) 
a'i  \  F  ix)  . 


a 

fts ;      a; 
:  ay ;      a] 


^  ,  F  ix)  =  «5 
J?'  (a;)  =  a, 
F  ix)  =  a, 

=  fffi 

-=  "7   J 


.(27) 


exa.ivif>i>e;s. 

26.  a;'—  6  ic«  +  23  x*—  49  ar!  +  225  x^  —  120  ,r  —  200  =0. 

In  this  solution  we  shall  begin  at  the  top  and  bottom  and  work  to  the  center.     Hence,  we 
make  «,  =  6  ,  l^—  ~4,  ?i'  =  «, ,  I'^  —  a^  ,   f,"  =  «," ,  l'^  =  a'^  ,  «[■  =  «  r=  .8  ,  say. 


6 
5 
3 
.8 

—  1.6 

—  2.8 

—  4 


Fix) =  5.2465  =  a, 

Fzix)^         3+;       S  \  F   ix) =  2.8284  =a2 

F^  ix)  =      2.4 ;        2.4  I  F^  ix)  =  1.5  ;    1.5  |  F  ix)  =  1.4231  =  a^ 
Ft  ix)  ==         .5 ;  .5  I  J'a  ix)  =  i  a. 

Ft  ix)=—    .5 ;  —    .5  I  F^  ix)  =  i  a, 

F^ix)^  —1.6;  —1.6  \  F    ix)   =—    .6696  =  a, 

Fix)=    =  —  2.8284=0, 


18  DIRECT    METHOD    OF    FINDING    THE    APPROXIMATE    VALUES 

It  appears  that  «<  and  ffj  =  i ,  but  in  reality  «4  =  —  .6696,  and  a,  and  a,  =  i.     See  (22). 

27.  x'  —  6  a;»  +  17  a;  —  29  =  0. 

Here,  1^  — -{-  o  ,  /^  ==  —  4,  and  /"  (.r)  =  21  x-'  —  60  .t'.  Hence  the  roots  of  /"  (.r)  =  0  , 
are  ].9,  0,  0,  0,  —  1.9.  Therefore,  there  being  no  equal  roots,  the  only  possible  real  roots  are 
0  I  F(.r),     1.9  i  l-'Ca;),    0  |  F  (x),   —  1.9  |  F(w),    and  —  4  |  F  (x). 


5  I  F  ix)  =  2.4  + 
1.9  \  Fix)=  i 


0  I  F  (X)  =       i 
1.9  I  F(x)  =  —  1.9- 


i  \  F(x)  =  —  2.1 


EQUATIONS  OF  THE  Nth  DEGREE. 

31.  It  is  evident  that  the  process  may  be  applied  to  any  numerical  equation. 

ROOTS  OF  NUMBERS. 

32.  In  extracting  the  roots  of  numbers  by  formula  (1),  the  approximation  is  very  rapid. 

EXAMPLES. 

28.  Find  the  cube  root  of 60. 

Make  (a;  —  3  )'  =  30 ;  whence  af —  9  x^  +  27  a;  —  57  =  0. 

(  (9  a;  — 54)a;  +  171)  a; 


Substituting  in  (1),  making  «  =  1,  ^  (x)  -. 


(.«2  _  27)  a;  +  114 


6  I  ^  (a')  I  3  +  v^  =  6.107  + 
6.107  I        "  =6.1072325050. 

•••  1/30  =3  3.10723250.50. 

29.    Find  the  square  root  of  3. 

Make  {x  —  2)^=  3  ;    whence  .-r?  —  4  .r  -j-  1  =:  0 ;    and  substituting  in  (1),   making  .f  =  1, 

~^^   '        (X  +  1)  (.r  -  1) 

4  I  ^  (a,')  I  2  +  ^/J=  3.73  + 
3.73  I       "  =  3.7320508  + 

3.7320508  |       "  =3.732050807568877293527 

.-.  ,/y=  1.7320,50807568877293527, 

These  results  are  correct  as  far  as  carried, 


OF    THE    RKAI,     KOOTS    OK     NUMEKICAI.    KQUATIO^^S.  19 


NOTES. 

1.  We  have  made  no  attempt  to  estimate  the  rapidity  of  the  approximation,  because  the  method  of 
doing  so  is  given  in  many  of  our  best  text-books.  See  Todhunter's  Theory  of  Equations,  page  1^9.  Suffice  to  say 
that  "the  number  of  exact  decimal  places  is  nearly  doubled  at  each  step." 

2.  In  all  the  practical  applications  of  equations  a  greater  degree  of  accuracy  in  the  roots  than  three  or  four 
places  of  decimals  is  seldom,  if  ever,  required. 

3.  In  order  to  avoid  the  possibility  of  an  error  in  seeking  a  root,  the  number  to  be  substituted  for  x  should 
be  a  little  larger  or  smaller  than  the  partial  result  of  the  last  substitution,  according  as  the  root  sought  is  less  or 
greater  than  that  result,  respectively.  This  is  especially  necessary  when  two  roots  are  very  nearly  equal 
to  each  other. 

CONCLUSION, 

The  following  is  an  extract  from  "  The  Teaching  and  History  of  Mathematics  in  the  United  States,"  page 
363,  by  Prof.  Florian  Cajori,  M.  S.: 

"  In  this  connection  a  quotation  from  Gerhardt's  Geschichle  der  Mutheinatik  in  DeulscUland  (p.  205)  is 
instructive. 

"Savshe:  'The  advocates  of  the  combinatorial  analysis  were  of  the  opinion  that  with  the  complete 
solution  of  this  problem  (of  reversion  of  series)  was  given  als  >  the  general  solution  of  equations.  But  here 
they  overlooked  an  important  point — the  convergency  or  divergency  of  the  series  which  was  obtained  for  the 
value  of  the  unknown  quantity.  Modern  analysis  justly  demanded  an  investigation  of  this  point,  inasmuch  as 
the  usefulness  of  the  results  i.s  completely  dependent  upon  it.'  It  thus  appears  that,  through  the  misuse  of 
infinite  series,  the  Germans  were  temporarily  led  to  believe  that  they  had  reached  a  result  which  mathematicians 
had  so  long  but  vainly  striven  to  attain,  namely,  the  algebraic  solution  of  equations  higher  than  the  fourth 
degree.  It  will  be  observed  that  their  method  lacked  generality,  since  it  could  at  best  not  yield  more  than  one 
root  of  an  equation.  But  in  the  determination  of  this  one  root  the  combinatorial  school  was  deceived.  The 
result  was  a  mere  delusion — a  mirage  produced  by  the  refraction  of  the  rays  of  reasoning  from  their  true  path 
while  passing  through  the  atmosphere  of  divergent  f-eries." 

The  "general  solution"  referred  to  above  is  a  direct  and  absolute  method  of  finding  the  approximate 
values  of  the  real  roots  of  numerical  equations  of  the  higher  degrees.  Therefore  I  claim,  I  trust  without  immodesty, 
to  have  "  reached  a  result  which  mathematicians  have  so  long  but  vainly  striven  to  attain." 


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